Is it possible to share GLUquadrics between different contexts (as you can do with displaylists, textures, shaders and so on…) using wglShareLists?
Probably not, as it is GLU not GL. I never heard of something like that.
You could try to share it without wglShareLists. IMHO glu uses the immediate mode (glBegin(???)…glEnd()) or maybe vertexarrays…
Most times the glu quadrics are slow, own vbo based solutions are often much faster.